As a rule, gerunds should be used like their verbal counterparts. If the verb is a phrasal verb, then the gerund should be constructed accordingly.
So your example "Thinking of you makes me happy" is correct, because you'd use "of" if you rephrased this with "think" as a verb:
When I think of you, it makes me happy.
However, this is incorrect
He has finished washing of the dishes.
because you'd say "I don't want to wash the dishes," not "I don't want to wash of the dishes." That last sentence makes no sense in English.
The Inevitable and Rarely Used Exception
You knew there was going to be an exception, right?
In a few cases, using the gerund + of is correct when it is important to emphasize the specific act the gerund describes. It also sounds extremely formal to an English speaker; it is not a construction one would use in everyday speech except as a joke.
Because this construction refers to a definite act, it must take the definite article "the." The phrase "She enjoys reading of poems" would sound odd to an English speaker, but "She enjoys the reading of poems" is perfectly correct, though having an elevated tone.
And because "the reading of poems" draws attention to the act and not the person performing the act, it can imply the reading of poems by others in addition to her own reading of poems.
A few more examples:
The signing of the Declaration of Independence actually took place on July 2.
The naming of new Nobel Prize laureates is always exciting.
I love the turning of the seasons.
Notice how the use of "the" refers to a specific act or event. The sentence
Signing the Declaration of Independence actually took place on July 2.
does not make sense because without "the," it implies a habitual action, not a specific act or event.
Humorous use of this construction usually makes an insignificant event sound more important than it really is:
Now is the time for the washing of dishes.
I humbly invite you to join me in the drinking of beer.
These are very similar in meaning, almost identical.
And now the waiting begins.
^ The speaker is here saying that an ongoing action of waiting is starting.
And now the wait begins.
^ The speaker is here saying that a single event that occurs over a time interval - the wait - is starting.
In particular, the choice is just slightly connotative, and almost completely stylistic. There is absolutely no difference in time period signaled; you're overthinking the use with the "duration vs. frequency" in your question. The two sentences are very nearly exactly the same. Perhaps the first can focus the listener more on the person or people doing the waiting, while the second is slightly more "passive" or removed from the actors. Perhaps not.
Best Answer
Strictly speaking, the sentence is incorrect because listening of playlists is ungrammatical (unless the playlists are doing the listening). Since categorization takes of and editing takes of but listening takes to, you have to write this:
This is grammatically correct but it sounds very clumsy. If each noun took the same preposition, you could use the same preposition for all three:
This is grammatically correct but it sounds even clumsier because of the repetition of play. People would rather make a subtle grammatical error than write a sentence that sounds this clumsy.
The fact that the first two of the nouns take of probably led people to ignore the incorrect listen of for almost ten years now.
Another “fudge” solution is to choose the preposition to agree with only the nearest noun even if it disagrees with all the others, known as “proximate agreement”:
There is, however, a better way:
This is clearer because the users are mentioned explicitly, and the nominalized verbs are replaced with plain old infinitive verbs. The preposition to only agrees with listen, but that's OK: categorize and edit are transitive verbs, which take an object without any preposition at all. So, to connects only with listen and there is no disagreement with categorize and edit. So, this version has perfect grammar as well as greater clarity.
By the way, many gerunds do take of. For example: editing of playlists, feeding of animals, planting of gardens, singing of songs, etc. Also, gerunds normally function as nouns. In the original sentence, editing and listening are objects of allow, just like categorization.